Brackets [ ] and braces { } have the same function as parentheses. They are all grouping symbols. After parentheses are used, then for clarity we use brackets. After brackets, braces.

Removing brackets or braces will follow the same rules for removing parentheses.

Example 2.a − [b − (c − d + e)]

We will remove all the grouping symbols. We will do it by removing the
brackets first. Then we will do it again removing the parentheses first. The student should have the skill to do it either way.

So, upon removing the brackets:

a − [b − (c − d + e)]

=

a − b + (c − d + e).

Within the brackets, there are two terms. The first term is b. The second term is −(c − d + e). (See Problem 1c above.) Since the brackets are preceded by − , the sign of each of the two terms changes. The signs within the term (c − d + e) do not change.

Finally, we remove the parentheses, which are preceded by + :

=

a − b + c − d + e.

Now let us do this same problem by removing the parentheses first:

a − [b − (c − d + e)]

=

a − [b − c + d − e]

=

a − b + c − d + e.

Since the parentheses are preceded by − , every sign within them changes. And since the brackets also are preceded by − , every sign within them changes.

Problem 17.

a) First remove the brackets, then remove the parentheses.

w + [x − (y + z)] = w + x − (y + z) = w + x − y − z.

First remove the parentheses, then remove the brackets.

w + [x − (y + z)] = w + [x − y − z] = w + x − y − z.

b) First remove the brackets, then remove the parentheses.

w − [x + (y − z)] = w − x − (y − z) = w − x − y + z.

First remove the parentheses, then remove the brackets.

w − [x + (y − z)] = w − [x + y − z] = w − x − y + z.

c) First remove the brackets, then remove the parentheses.

w − [x − (y + z)] = w − x + (y + z) = w − x + y + z.

First remove the parentheses, then remove the brackets.

w − [x − (y + z)] = w − [x − y − z] = w − x + y + z.

d) First remove the brackets, then remove the parentheses.

w + [x − (y − z)] = w + x − (y − z) = w + x − y + z.

First remove the parentheses, then remove the brackets.

w + [x − (y − z)] = w + [x − y + z] = w + x − y + z.

Problem 18. Remove all the grouping symbols. Simplify as you go by evaluating the numbers. Remove the brackets first.

a) 5 − [3 − (x − 2)]

=

5 − 3 + (x − 2)

=

2 + x − 2

=

x.

b) 5 − [3 − (x + 2)]

=

5 − 3 + (x + 2)

=

2 + x + 2

=

x + 4.

c) −5 + [3 − (x − 2)]

=

−5 + 3 − (x − 2)

=

−2 − x + 2

=

−x.

d) 5 − [−3 − (x + 2)]

=

5 + 3 + (x + 2)

=

8 + x + 2

=

x + 10.

Problem 19.

a) First remove the braces, then the brackets, then the parentheses.a) Simplify by adding the numbers.